Example Question - hypotenuse length
Here are examples of questions we've helped users solve.
Solving for the Length of a Side in a Right-Angled Triangle
The image shows a right-angled triangle, which means we can use the Pythagorean theorem to solve for the missing side, labeled as "x." The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In the given triangle, we have one side that is 9 cm and another side that is 12 cm. Assuming that the 12 cm side is the hypotenuse (since it's opposite what is presumably the right angle), we can compute the length of side "x" using the following equation: \( c^2 = a^2 + b^2 \) Here, \( c \) represents the hypotenuse, while \( a \) and \( b \) represent the other two sides. Rearranging the equation to solve for \( a \), we get: \( a^2 = c^2 - b^2 \) Let's substitute the given values: \( x^2 = 12^2 - 9^2 \) \( x^2 = 144 - 81 \) \( x^2 = 63 \) Taking the square root of both sides to solve for \( x \), we find: \( x = \sqrt{63} \) The square root of 63 can be simplified further by factoring out perfect squares. \( 63 = 9 \times 7 \), and since \( 9 \) is a perfect square, we get: \( x = \sqrt{9 \times 7} \) \( x = \sqrt{9} \times \sqrt{7} \) \( x = 3\sqrt{7} \) Therefore, the length of side \( x \) is \( 3\sqrt{7} \) cm, which cannot be simplified further without a decimal approximation. If a decimal approximation is needed, \( \sqrt{7} \approx 2.645 \), and multiplying this by 3 gives: \( x \approx 3 \times 2.645 \) \( x \approx 7.935 \) So the side "x" is approximately 7.935 cm long.
Solving for Hypotenuse Length in a Right Triangle
To solve for the length of the hypotenuse in a right triangle, you can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The Pythagorean theorem is written as: \[ c^2 = a^2 + b^2 \] According to the image, one leg (a) is 36 km long, and the other leg (b) is 77 km long. We are solving for the hypotenuse (c). Plugging the values into the theorem: \[ c^2 = (36)^2 + (77)^2 \] \[ c^2 = 1296 + 5929 \] \[ c^2 = 7225 \] Now, take the square root of both sides to solve for c: \[ c = \sqrt{7225} \] \[ c = 85 \] Therefore, the length of the hypotenuse is 85 km. Since the question did not ask for rounding unless necessary and we got an exact whole number, we don't round the answer. It is exactly 85 kilometers.
Solving for the Length of Legs in a 45-45-90 Triangle
The image shows a right-angle triangle with a 45-degree angle. Since the angles in a triangle add up to 180 degrees and we already have a right angle (90 degrees) and one 45-degree angle, the other angle must also be 45 degrees. This makes the triangle a 45-45-90 triangle, which is a form of an isosceles right triangle. In a 45-45-90 triangle, the lengths of the legs (the two shorter sides) are equal, and the length of the hypotenuse (the longest side, opposite the right angle) is √2 times the length of a leg. The figure gives the hypotenuse as length 3. Therefore, to find the length of each leg (let's call it 'L'), we can use the proportion that L is to 3 (the hypotenuse) as 1 is to √2: L / 3 = 1 / √2 Multiplying both sides by 3 to solve for L: L = 3 / √2 To rationalize the denominator: L = (3 / √2) * (√2 / √2) L = (3√2) / 2 So, the length of each leg of the triangle is (3√2) / 2 units.
Solving for Leg Lengths in a 45-45-90 Triangle
The image shows a right triangle with one angle of 45 degrees and the hypotenuse having a length of 3 units. Given that the triangle is a right triangle with a 45-degree angle, we can infer that this is a 45-45-90 triangle, which is an isosceles right triangle. In such triangles, the lengths of the legs are equal, and the length of the hypotenuse is √2 times the length of each leg. Let's use the properties of a 45-45-90 triangle to find the lengths of the two legs. If \( a \) is the length of one leg, we have: \( a \cdot \sqrt{2} = 3 \) Now we solve for \( a \): \( a = \frac{3}{\sqrt{2}} \) To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{2} \): \( a = \frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \) \( a = \frac{3 \cdot \sqrt{2}}{2} \) So both legs of the triangle are \( \frac{3 \cdot \sqrt{2}}{2} \) units long.
Solving the Length of the Hypotenuse in an Isosceles Right Triangle
The image shows a right triangle with one angle marked as 45 degrees, which means it is an isosceles right triangle (since the other non-right angle must also be 45 degrees). In such triangles, the legs are congruent. If the triangle is labeled with points G, J, and H, where GH is the hypotenuse, and the leg GJ is labeled as "4√2," then we can find the length of GH using the Pythagorean theorem. However, for an isosceles right triangle, we have a simpler relationship: In an isosceles right triangle, the length of the hypotenuse is √2 times the length of a leg. Given GJ = 4√2, the length of GH (the hypotenuse) is: GH = GJ * √2 GH = 4√2 * √2 GH = 4 * (√2 * √2) GH = 4 * 2 GH = 8 Therefore, the exact value of GH is 8, which corresponds to option C.
Pythagorean Theorem Application in Vacation Center Design
D'accord, examinons l'image. Nous avons un dessin représentant le rez-de-chaussée d'un centre de vacances composé de deux pièces. Pour résoudre ce problème, nous allons avoir besoin d'utiliser le théorème de Pythagore, qui s'applique dans un triangle rectangle et qui stipule que la somme des carrés des longueurs des deux côtés adjacents à l'angle droit est égale au carré de la longueur de l'hypoténuse. Dans le cas de ce problème, nous pouvons voir que la pièce où se trouve le "coin jeu" est un carré parce que les côtés AE et EB ont la même longueur de 20 mètres. Cela signifie que le triangle DEC est rectangle en E. La longueur DE est le côté du carré, donc DE = AE = 20 m. La longueur DC est la somme des côtés des deux pièces, donc DC = BC + AB = 20 m + 25 m = 45 m. Nous cherchons à vérifier la longueur de l'hypoténuse, c'est-à-dire EC. Selon le théorème de Pythagore: EC² = DE² + DC² Remplaçons par les valeurs connues: EC² = 20² + 45² EC² = 400 + 2025 EC² = 2425 Pour trouver la longueur de EC, nous prenons la racine carrée des deux côtés: EC = √2425 Si EC doit être égale à 15 m, vérifions-le: 15² = 225 Nous voyons que 225 est différent de 2425. Donc, la longueur EC ne peut pas être de 15 m. En prenant la racine carrée de 2425, nous obtenons la longueur réelle de EC. EC ≈ √2425 EC ≈ 49,25 m (approximativement) La longueur réelle de EC est donc approximativement de 49,25 mètres. Il y a une erreur dans l'énoncé si on affirme que EC doit être de 15 mètres car cela ne correspond pas aux calculs selon le théorème de Pythagore.
Calculating the Length of a Hypotenuse
The Pythagorean theorem states that for a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The theorem can be written as: c² = a² + b² In the provided right-angled triangle, side a is 9 ft and side b is 12 ft. You are asked to find the length of the hypotenuse (c). Using the Pythagorean theorem: c² = 9² + 12² c² = 81 + 144 c² = 225 To find the length of c, take the square root of both sides: c = √225 c = 15 ft Therefore, the length of the hypotenuse is 15 feet.
Solving Right-Angled Triangle Problems with the Pythagorean Theorem
The image shows two separate mathematical problems involving right-angled triangles, one labeled as question 4 and the other as question 5. Both problems are about finding the length of the missing side of the triangle, which is a classic Pythagorean theorem problem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. For question 4, the sides of the triangle are given as 3 (one leg) and 4 (the other leg), and we are to find the hypotenuse. By applying the Pythagorean theorem (a^2 + b^2 = c^2), where 'c' is the hypotenuse and 'a' and 'b' are the other sides, we can solve the triangle: 3^2 + 4^2 = c^2 9 + 16 = c^2 25 = c^2 c = √25 c = 5 So, the length of the hypotenuse is 5 and the correct answer is not shown as marked in the image. For question 5, the sides of the triangle are given as 7 (one leg) and 24 (the other leg), and we are to find the hypotenuse. We apply the Pythagorean theorem again: 7^2 + 24^2 = c^2 49 + 576 = c^2 625 = c^2 c = √625 c = 25 The length of the hypotenuse for question 5 is therefore 25, which corresponds with answer choice B. To summarize: - Question 4: The hypotenuse = 5 (not in the choices) - Question 5: The hypotenuse = 25 (choice B)
Solving for Length of Hypotenuse in Right-Angled Triangle
You have presented a right-angled triangle ABC with an altitude BD from the right angle B to the hypotenuse AC, creating two smaller right-angled triangles, ABD and BDC. You have provided several pieces of information: 1. The angle \( \angle ABC = 15^\circ \) 2. The angle \( \angle BAD = 45^\circ \) 3. The length of BD is equal to that of DC, i.e., \( BD = DC \) You've asked to solve for the value of y, where y is the length of AC. But there's no direct indication of how the lengths on the triangle relate to y. The given angles and conditions suggest that we can calculate the length of AC using trigonometric ratios. Since triangle ABD and BDC are both isosceles and right-angled, we know several relationships: 1. \( AB = BD \) (since \( \angle BAD = 45^\circ \) it is an isosceles right-angled triangle, so the two non-hypotenuse sides are equal) 2. \( BD = DC \) (given) 3. Therefore, \( AB = BD = BC \), and those lengths are all equal to each other. To find y, we need to use these relationships and the fact that \( \angle ABC = 15^\circ \). We'll use the trigonometric identity for the sine of a 15-degree angle, as we're given \( \angle ABC \) and we have a right-angled triangle. In right-angled triangle ABC: \[ \sin(15^\circ) = \frac{AB}{AC} = \frac{BD}{AC} \] We know that \( \sin(15^\circ) \) can be expressed using the half-angle identity: \[ \sin(15^\circ) = \sin(30^\circ / 2) = \sqrt{\frac{1 - \cos(30^\circ)}{2}} = \sqrt{\frac{1 - \sqrt{3}/2}{2}} \] Simplifying the expression under the square root gives us: \[ \sin(15^\circ) = \sqrt{\frac{2 - \sqrt{3}}{4}} \] Using this identity, we can relate the length of AC to BD: \[ \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{BD}{AC} \] Since \( BD = BC \) and \( BC = AB \), let's denote the length of BD (also AB and BC) as x. Then the above equation becomes: \[ \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{x}{y} \] We can solve for y by multiplying both sides by y and dividing by \( \sqrt{\frac{2 - \sqrt{3}}{4}} \): \[ y = \frac{x}{\sqrt{\frac{2 - \sqrt{3}}{4}}} \] However, we have not been provided the actual numerical length for x (BD, AB, or BC), so we cannot provide a numerical solution for y without additional information. If you have a length for BD, AB, or BC, you can substitute that value into the equation to solve for y.
Solving for the Length of a Side in a Right-Angled Triangle
The image shows a right-angled triangle with sides labeled. The hypotenuse is labeled as 73, one leg is labeled as 48, and the other leg, which is opposite the right angle, is labeled with the variable N. To solve for N, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The Pythagorean theorem formula is as follows: c² = a² + b² In this case: c = 73 (the hypotenuse), a = 48 (one of the other sides), b = N (the side we're trying to find). Plugging the numbers we know into the formula gives us: 73² = 48² + N² Now calculate each term: 73² = 5329, 48² = 2304, And putting them into the equation: 5329 = 2304 + N² Next, subtract 2304 from both sides to solve for N²: 5329 - 2304 = N² 3025 = N² Finally, take the square root of both sides to find N: N = √3025 N = 55 Therefore, the value of N is 55.
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